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Lagrangian structure functions in turbulence: A quantitative comparisonbetween experiment and direct numerical simulation
L. Biferale,1 E. Bodenschatz,2,a� M. Cencini,3 A. S. Lanotte,4,a� N. T. Ouellette,5,b� F. Toschi,6
and H. Xu5
1International Collaboration for Turbulence Research and Dipartimento Fisica and INFN, Università di“Tor Vergata,” Via della Ricerca Scientifica 1, 00133 Roma, Italy2International Collaboration for Turbulence Research, Max Planck Institute for Dynamicsand Self-Organization, Am Fassberg 17, D-37077 Goettingen, Germany and Laboratory of Atomicand Solid-State Physics, Cornell University, Ithaca, New York 14853, USA3International Collaboration for Turbulence Research, INFM-CNR, SMC Dipartimento di Fisica, Universitàdi Roma “La Sapienza,” p.zle A. Moro 2, 00185 Roma, Italy and Istituto dei Sistemi Complessi-CNR,via dei Taurini 19, 00185 Roma, Italy4International Collaboration for Turbulence Research, CNR-ISAC, Sezione di Roma, Via Fosso delCavaliere 100, 00133 Roma, Italy and INFN, Sezione di Lecce, 73100 Lecce, Italy5International Collaboration for Turbulence Research and Max Planck Institute for Dynamicsand Self-Organization, Am Fassberg 17, D-37077 Goettingen, Germany6International Collaboration for Turbulence Research and Istituto per le Applicazioni del Calcolo CNR,Viale del Policlinico 137, 00161 Roma, Italy and INFN, Sezione di Ferrara, Via G. Saragat 1,I-44100 Ferrara, Italy
�Received 2 August 2007; accepted 8 April 2008; published online 5 June 2008�
A detailed comparison between data from experimental measurements and numerical simulations ofLagrangian velocity structure functions in turbulence is presented. Experimental data, at Reynoldsnumber ranging from R�=350 to R�=815, are obtained in a swirling water flow betweencounter-rotating baffled disks. Direct numerical simulations �DNS� data, up to R�=284, are obtainedfrom a statistically homogeneous and isotropic turbulent flow. By integrating information fromexperiments and numerics, a quantitative understanding of the velocity scaling properties over awide range of time scales and Reynolds numbers is achieved. To this purpose, we discuss in detailthe importance of statistical errors, anisotropy effects, and finite volume and filter effects, finitetrajectory lengths. The local scaling properties of the Lagrangian velocity increments in the two datasets are in good quantitative agreement for all time lags, showing a degree of intermittency thatchanges if measured close to the Kolmogorov time scales or at larger time lags. This systematicstudy resolves apparent disagreement between observed experimental and numerical scalingproperties. © 2008 American Institute of Physics. �DOI: 10.1063/1.2930672�
I. INTRODUCTION
Understanding the statistical properties of a fully devel-oped turbulent velocity field from the Lagrangian point ofview is a challenging theoretical and experimental problem.It is a key ingredient for the development of stochastic mod-els for turbulent transport in such diverse contexts as com-bustion, pollutant dispersion, cloud formation, and industrialmixing.1–4 Progress has been hindered primarily by the pres-ence of a wide range of dynamical time scales, an inherentproperty of fully developed turbulence. Indeed, for a com-plete description of particle statistics, it is necessary to fol-low their paths with very fine spatial and temporal reso-lution, on the order of the Kolmogorov length and timescales � and ��. Moreover, the trajectories should be trackedfor long times, on the order of the eddy turnover time TL,requiring access to a vast experimental measurement region.The ratio of the above time scales can be estimated asTL /���R�, and the Taylor microscale Reynolds number R�
ranges from hundreds to thousands in typical laboratory ex-periments. Despite these difficulties, many experimental andnumerical studies of Lagrangian turbulence have been re-ported over the years.5–33 Here, we present a detailed com-parison between state-of-the-art experimental and numericalstudies of high Reynolds number Lagrangian turbulence. Wefocus on single-particle statistics, with time lags rangingfrom smaller than �� to order TL. In particular, we study theLagrangian velocity structure functions �LVSFs�, defined as
Sp��� = ����v�p� = ��v�t + �� − v�t��p� , �1�
where v denotes a single velocity component.In the past, the corresponding Eulerian quantities, i.e.,
the moments of the spatial velocity increments, have at-tracted significant interest in theory, experiments, and nu-merical studies �for a review, see Ref. 34�. It is now widelyaccepted that spatial velocity fluctuations are intermittent inthe inertial range of scales, for ��r�L, L being the largestscale of the flow. By intermittency, we mean anomalous scal-ing of the moments of the velocity increments, correspond-ing to a lack of self-similarity of their probability densityfunctions �PDFs� at different scales. In an attempt to explain
a�Authors to whom correspondence should be addressed. Electronic mail:[email protected], [email protected].
b�Present address: Department of Physics, Haverford College, PA 19041.
PHYSICS OF FLUIDS 20, 065103 �2008�
1070-6631/2008/20�6�/065103/12/$23.00 © 2008 American Institute of Physics20, 065103-1
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Eulerian intermittency, many phenomenological theorieshave been proposed, either based on stochastic cascade mod-els �e.g., multifractal descriptions35–37� or on closures of theNavier–Stokes equations.38 Common to all these models isthe presence of nontrivial physics at the dissipative scale, r��, introduced by the complex matching of the wild fluc-tuations in the inertial range and the dissipative smoothingmechanism at small scales.39,40 Numerical and experimentalobservations show that clean scaling behavior for the Eule-rian structure functions is found only in a range 10��r�L �see Ref. 41 for a collection of experimental andnumerical results�. For spatial scales r�10�, multiscalingproperties, typical of the intermediate dissipative range, areobserved due to the superposition of inertial and dissipativerange physics.40
Similar questions can be raised in the Lagrangian frame-work: �i� Is there intermittency in Lagrangian statistics? �ii�Is there a range of time lags where clean scaling properties�i.e., power law behavior� can be detected? �iii� Are theresignatures of the complex interplay between inertial and dis-sipative effects for small time lags ��O����?
In this paper, we shall address the above questions bycomparing direct numerical simulations �DNSs� and labora-tory experiments. Unlike Eulerian turbulence, the study ofwhich has attracted experimental, numerical, and theoreticalefforts since the past 30 years, Lagrangian studies becomeavailable only very recently mainly due to the severe diffi-culty of obtaining accurate experimental and numerical dataat sufficiently high Reynolds numbers. Consequently, the un-derstanding of Lagrangian statistics is still poor. This ex-plains the absence of consensus on the scaling properties ofthe LVSF. In particular, there have been different assess-ments of the scaling behavior,
Sp��� = ����v�p� � ���p�, �2�
when a single number, i.e., the scaling exponent ��p�, is ex-tracted over a range of time lags.
Measurements using acoustic techniques10,15 gave thefirst values of the exponents ��p�, measuring scaling proper-ties in the range 10�����TL. Subsequently, experimentsbased on complementary metal oxide semiconductor�CMOS� sensors26,28 provided access to scaling propertiesfor shorter time lags, 2�����6��, finding more intermittentvalues, although compatible with Ref. 10. DNS data, ob-tained at lower Reynolds number, allowed simultaneousmeasurements in both of these ranges.23,29 For 10�����50��, scaling exponents were found to be slightly less in-termittent than those measured with the acoustic techniques,although again compatible within error bars. On the otherhand, DNS data29,33,42 for small time lags, 2�����6��,agree with scaling exponents measured in Ref. 26.
The primary goal of this paper is to critically comparestate-of-the-art numerical and experimental data in order toanalyze intermittency at both short, ����, and intermediate,�����TL, time lags. This is a necessary step both to bringLagrangian turbulence up to the same scientific standards asEulerian turbulence and to resolve the conflict between ex-periment and simulations �see also Refs. 33, 42, and 43�.
To illustrate some of the difficulties discussed above, in
Fig. 1, we show a compilation of experimental and numericalresults for the second-order Lagrangian structure function atvarious Reynolds numbers �see later for details�. Here and inthe following, we consider the LVSF averaged over the threecomponents, so that expression �1� for moment of order p isgeneralized to Sp���= � 1
3�i����vi�p�, and the index i runs over
the spatial components. The curves are compensated with thedimensional prediction given by the classical Kolmogorovtheory in the inertial range,44 S2���=C0��, where � is theturbulent kinetic energy dissipation. The absence of any ex-tended plateau and the trend with the Reynolds number in-dicate that the inertial range, if any, has not developed yet.The same trends have been observed in other DNS studies27
and by analyzing the temporal behavior of signals with agiven power law Fourier spectrum.45
We stress that assessing the actual scaling behavior ofthe second �and higher� order LVSFs is crucial for the devel-opment of stochastic models for Lagrangian particle evolu-tion. Indeed, these models are based on the requirement thatthe second-order LVSF scales as S2�����. The issues ofwhether the predicted scaling is ever reached and ultimatelyhow the LVSF deviate as a function of the Reynolds numbersremain to be clarified.
Moreover, an assessment of the presence of Lagrangianintermittency calls for more general questions about phe-nomenological modeling. For instance, multifractal modelsderived from Eulerian statistics can be easily translated to theLagrangian framework10,23,46,47 with some degree ofsuccess.10,13,18
The material is organized as follows. In Sec. II, we de-scribe the properties of the experimental setup and the DNSs,detailing the limitations in both sets of data. A comparison ofLVSFs is considered in Sec. III. Section III A presents adetailed scale-by-scale discussion of the local scaling expo-nents, which is the central result of the paper. Section IV
0
1
2
3
4
5
6
7
10-2 10-1 100 101 102
S2(
τ)/ε
τ
τ/τη
DNS1DNS2EXP1EXP2EXP3EXP4
FIG. 1. Log-log plot of the second-order LVSF �averaged over the threecomponents� normalized with the dimensional prediction, i.e., S2��� / ����, atvarious Reynolds numbers and for all data sets. Details can be found inTables I and II. EXP2 and EXP4 refer to experiments at the same Reynoldsnumber �R�=690�, but with different measurement volumes �larger inEXP4�; in particular, EXP2 and EXP4 better resolve the small and largetime lag ranges, respectively, and intersect for � /���2. We indicate with asolid line the resulting data set made of data from EXP2 �for � /���2� andEXP4 �for � /��2�; a good overlap among these data is observed in therange 2�� /���8. For all data sets, an extended plateau is absent, indicat-ing that the power law regime typical of the inertial range has not yet beenachieved, even at the highest Reynolds number, R��815, in experiment.
065103-2 Biferale et al. Phys. Fluids 20, 065103 �2008�
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draws conclusions and offers perspectives for the futurestudy of Lagrangian turbulence.
II. EXPERIMENTS AND NUMERICAL SIMULATIONS
Before describing the experimental setup and the DNS,we shall briefly list the possible sources of uncertainties inboth experimental and DNS data. In general, this is not aneasy task. First, it is important to discern the deterministicfrom the statistical sources of errors. Second, we must beable to assess the quantitative importance of both types ofuncertainties on different observables.
Deterministic uncertainties. For simplicity, we report inthis work the data averaged over all three components of thevelocity for both the experiments and the DNS. Since neitherflows in the experiments nor the DNS are perfectly isotropic,a part of the uncertainty in the reported data comes from theanisotropy. In the experiments, the anisotropy reflects thegeneration of the flow and the geometry of the experimentalapparatus. The anisotropy in DNS is introduced by the finitevolume and by the choice of the forcing mechanism. In gen-eral, the DNS data are quite close to statistical isotropy, andanisotropy effects are appreciable primarily at large scales.This is also true for the data from the experiment, especiallyat the higher Reynolds numbers. An important limitation ofthe experimental data is that the particle trajectories havefinite length due both to finite measurement volumes and tothe tracking algorithm, which primarily affect the data forlarge time lags. It needs to be stressed, however, that in thepresent experimental setup due to the fact that the flow is notdriven by bulk forces, but by inertial and viscous forces atthe blades, the observation volume would anyhow be limitedby the mean velocity and the time it takes for a fluid particleto return to the driving blades. At the blades, the turbulenceis strongly influenced by the driving mechanism. Therefore,in the experiments reported here, the observation volumewas selected to be sufficiently far away from the blades tominimize anisotropy. For short-time lags, the greatest experi-mental difficulties come from the finite spatial resolution ofthe camera and the optics, the image acquisition rate, datafiltering, and postprocessing, a step necessary to reducenoise. For DNS, typical sources of uncertainty at small timelags are due to the interpolation of the Eulerian velocity fieldto obtain the particle position, the integration scheme used tocalculate trajectories from the Eulerian data, and the numeri-cal precision of floating point arithmetic.
The statistical uncertainties for both the experimentaland DNS data arise primarily from the finite number of par-ticle trajectories and—especially for DNS—from the timeduration of the observation. We note that this problem is alsoreflected in a residual, large-scale anisotropy induced by thenonperfect averaging of the forcing fluctuations in the feweddy turnover times simulated. The number of independentflow realizations can also contribute to the statistical conver-gence of the data. While it is common to obtain experimentalmeasurements separated by many eddy turnover times, typi-cal DNS results contain data from at most a few eddy turn-over times.
We stress that, particularly for Lagrangian turbulence,
only an in-depth comparison of experimental and numericaldata will allow the quantitative assessment of uncertainties.For instance, as we shall see below, DNS data can be used toinvestigate some of the geometrical and statistical effects in-duced by the experimental apparatus and measurement tech-nique. This enables us to quantify the importance of some ofthe above mentioned sources of uncertainty directly. DNSdata are, however, limited to smaller Reynolds number thanexperiment; therefore, only data from experiments can helpto better quantify Reynolds number effects.
A. Experiments
The most comprehensive experimental data of Lagrang-ian statistics are obtained by optically tracking passive tracerparticles seeded in the fluid. Images of the tracer particles areanalyzed to determine their motion in the turbulent flow.6,7,48
Due to the rapid decrease of the Kolmogorov scale with Rey-nolds number in typical laboratory flows, previous experi-mental measurements were often limited to small Reynoldsnumbers.6,8 The Kolmogorov time scale at R��103 in alaboratory water flow was so far resolved only by using fourhigh speed silicon strip detectors originally developed forhigh-energy physics experiments.9,11 The one-dimensionalnature of the silicon strip detector, however, restricted thethree-dimensional tracking to a single particle at a time, lim-iting severely the rate of data collection. Recent advances inelectronics technology now allow simultaneous three-dimensional measurements of O�102� particles at a time byusing three cameras with two-dimensional CMOS sensors.High-resolution Lagrangian velocity statistics at Reynoldsnumbers comparable to those measured using silicon stripdetectors are therefore becoming available.26
Lagrangian statistics can also be measured acoustically.The acoustic technique measures the Doppler frequency shiftof ultrasound reflected from particles in the flow, which isdirectly proportional to their velocity.10,15 The size of theparticles needed for signal strength in the acoustic measure-ments can be significantly larger than the Kolmogorov scaleof the flow. Consequently, the particles do not follow themotion of fluid particles,11 and this makes the interpretationof the experimental data more difficult.15
The experimental data here presented are discussed inmuch detail in Refs. 26 and 28. In the following, we onlybriefly recall the main aspects of the experimental techniqueand data sets, whose parameters are summarized in Table I.Turbulence was generated in a swirling water flow betweencounter-rotating baffled disks in a cylindrical container. Theflow was seeded with polystyrene particles of size dp
=25 �m and density �p=1.06 g /cm3 that follow the flowfaithfully for R� up to 103.11 The particles were illuminatedby high-power Nd:YAG lasers, and three cameras at differentviewing angles were used to record the motion of the tracerparticles in the center of the apparatus. Images were pro-cessed to find particle positions in three-dimensional physi-cal space; the particles were then first tracked using a pre-dictive algorithm to obtain the Lagrangian trajectories.48 Dueto fluctuations in laser intensity, the uneven sensitivity of thephysical pixels in the camera sensor array, plus electronic
065103-3 Lagrangian structure functions in turbulence Phys. Fluids 20, 065103 �2008�
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and thermal noise, images of particles sometimes fluctuateand appear to blink. When the image intensity of a particlewas too low, the tracking algorithm lost that particle. Conse-quently, the trajectory of that particle was terminated. Whenthe image intensity is high again, the algorithm started a newtrajectory. The raw trajectories therefore contained manyshort segments that in reality belonged to the same trajectory.It is, however, possible to connect these segments by apply-ing a predictive algorithm in the six-dimensional space ofcoordinates and velocities. The trajectories discussed in thispaper were obtained with the latter method, which allows formuch longer tracks.
The Lagrangian velocities were calculated by smoothingthe measured positions and subsequently differentiating. AGaussian filter has been used to smooth the data. Smoothingand differentiation can be combined into one convolutionoperation by integration by parts; the convolution kernel issimply the derivative of the Gaussian smoothing filter.16 Thewidth of the Gaussian kernel was chosen to remove the noisein position measurements, but not to suppress the fluctua-tions, whose characteristic time scale is O���� or above. Thevelocity statistics have been found to be insensitive to thewidth of the Gaussian filter, provided that it is between�� /6 and �� /3 �see also below�. The temporal resolution ofthe camera system in the experiments reported here was suf-ficiently high to ensure that the fluctuations with time scalegreater than �� /6 were well resolved.
The uncertainty in position measurement, or the spatialresolution, is directly proportional to the size of the spatialdiscretization determined by the optical magnification and bythe size of the pixels on the CMOS sensor. Larger magnifi-cation gives better spatial resolution but also a smaller mea-surement volume. Indeed, the number of usable pixels of thecamera sensor array is fixed by the chip size and, at higherspeeds, by the imaging rate. The dynamic range of the cam-eras is not sufficient to cover the entire range of scales of theturbulence at the Reynolds numbers of interest. Therefore,two sets of experiments with different magnifications havebeen performed. The former set has a high spatial resolution
and focuses on the small scale quantities, although with arelatively small measurement volume �EXP1, 2, and 3 inTable I�. Then, in order to probe longer times and largerscales, the size of the measurement volume in the second setof measurements was chosen to be slightly smaller than theintegral scale �EXP4 in Table I�. In this data set, however, theuncertainty in position was larger and the short-time statisticswere severely affected. As a result, in order to have experi-mental data covering a wide range of time lags ������100��� at a given Reynolds number, one needs to mergedata from the two different experiments. This could be doneat R�=690 by using data from the small measurement vol-ume �EXP2� up to times ���6–7��� and using data fromlarge measurement volume �EXP4� at larger times. The pro-cedure is well justified as the two data sets match for inter-mediate time lags.
One noticeable difference between experiments and nu-merical simulations is the number of independent realiza-tions included in the statistics. While it is difficult to havemany statistically independent DNS results at one Reynoldsnumber, the experimental data usually contained O�103�records separated by a time interval of about 102TL. Each ofthese records lasted for �1–2�TL. The variation of the veloc-ity fluctuations calculated from the statistics of many recordsis shown in Fig. 2�a�. As it is clear from it, the three compo-nents do not fluctuate about the same value, indicating thepresence of anisotropy, which does not average away evenafter many eddy turnover times. These effects are introducedby the flow generation in the apparatus. In the following, theuncertainties in the data sets due to anisotropy were esti-mated by the difference between measurements made on dif-ferent components of the velocity field.
B. DNS
Nowadays, state-of-the-art numerics19,23,49,50 best suitedfor Eulerian statistics is able to reach Taylor scale Reynoldsnumbers of the order of R��1000 by using up to 40963
mesh points.50 Such extremely high Reynolds number DNS
TABLE I. Parameters of the experiments. Column 2 gives the Taylor microscale Reynolds numbers taken from previous experiments �Ref. 11� in the sameflow generating apparatus at the same driving speeds of the motors reported here. In Ref. 11, a small volume ��2 mm�3 at the center of the apparatus wasmeasured, while the measurement volumes in the current experiments were ��2 cm�3 and ��5 cm�3. We found that the local velocity fluctuations, measuredin subvolumes with size of �2 mm�3, varied by 5%–10%. This might be attributed to statistical convergence �typically �105 samples in each subvolume� andpossibly to the inhomogeneity of the flow. At the center of the apparatus, the fluctuation velocity was the highest and agreed with the value reported in Ref.11. The fluctuation velocities reported in column 3 are the spatial averages over the entire measurement volume. They are approximately 5% lower than thevalues given in Ref. 11. Since R�vrms�2 , a 5% difference in vrms� corresponds to a 10% difference in R�. Column 3 gives the value of the root-mean-squarevelocity fluctuations vrms� averaged over the three components. The integral length scale Lvrms�3 /�=7 cm was determined to be independent of Reynoldsnumber. TLL /vrms� is the eddy turnover time. Nf is the temporal resolution of the measurement in units of frames per ��. The measurement volume wasnearly a cube in the center of the tank and its linear dimensions are given in units of the integral length scale L. �x is the spatial discretization of the recordingsystem. The spatial uncertainty of the position measurements is roughly 0.1�x. NR is the number of independent realizations recorded �see text�. Ntr is thenumber of Lagrangian trajectories measured. We note that the energy dissipation rate was inferred from measurements of the second-order Eulerian structurefunctions.
No. R�
vrms��m/s�
��m2 /s3�
���m�
��
�ms�TL
�s�Nf
�f /���Measured volume
in �L3��x
��m /pix� NR Ntr
EXP1 350 0.11 2.0�10−2 84 7.0 0.63 35 0.4�0.4�0.4 50 500 9.3�105
EXP2 690 0.42 1.2 30 0.90 0.16 24 0.3�0.3�0.3 80 480 9.6�105
EXP3 815 0.59 3.0 23 0.54 0.11 15 0.3�0.3�0.3 80 500 1.7�106
EXP4 690 0.42 1.2 30 0.90 0.16 24 0.7�0.7�0.7 200 1200 6.0�106
065103-4 Biferale et al. Phys. Fluids 20, 065103 �2008�
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is, however, limited by the impossibility of integrating theflow for long time durations, due to the extremely high com-putational cost. In Lagrangian studies, it is necessary tohighly resolve the Eulerian velocity field to obtain preciseout-of-grid interpolation. The maximum achievable Rey-nolds number, on the fastest computers, is currently limitedto R��600 in order to accurately calculate the particle posi-tions and to achieve sufficiently long integrationtimes.4,19,23,27
Typically, such Lagrangian simulations last for a fewlarge-scale eddy turnover times, implying some unavoidableremaining anisotropy at large scales, even for nominally per-fectly isotropic forcing. The simulations analyzed here wereforced by fixing the total energy of the first two Fourier-space shells:51 E�k1�=�k��I1
�v�k�� and E�k2�=�k��I2�v�k��,
where I1= �0.5:1.5� and I2= �1.5:2.5� �the �k�=0 mode isfixed to zero to avoid a mean flow�. The three velocity com-ponents can instantaneously be quite different: when one ofthe three fluctuates, the others must compensate in order tokeep the total amplitude fixed �see, for instance, Fig. 2�b� fora visualization of this effect�. However, by averaging overmany eddy turnover times—when possible, as for the lower-resolution DNS shown in the inset of Fig. 2�b�—the forcingproduces a perfectly statistically isotropic flow. As the re-maining large-scale anisotropy is the main source of uncer-
tainty in the DNS results, we will estimate confidence inter-vals from the difference between the three components.
In the simulations, the main systematic error for smalltime lags comes from the interpolation of the Eulerian veloc-ity fields needed to integrate the equation for particle posi-tions,
X�t� = v�X�t�,t� . �3�
Of course, high-order interpolation schemes such as third-order Taylor series interpolation or cubic splines, now cur-rently used in parallel codes, partially remove this problem.27
If we compare DNS with the same value of kmax�, wherekmax is the maximum wavenumber resolved, cubic splinesgive higher interpolation accuracy. It has been reported52 thatcubic schemes may resolve the most intense events betterthan linear interpolation, especially for acceleration statistics;the effect, however, appears to be rather small especially asfar as velocity is concerned.
More crucial than the order of the interpolation schemeis the resolution of the Eulerian grid in terms of the Kolmog-orov length scale. To enlarge the inertial range as much aspossible, pure Eulerian simulations may not resolve thesmallest scale velocity fluctuations sufficiently well, bychoosing a grid spacing �x larger than the Kolmogorov scale�. Since this strategy may be particularly harmful to La-grangian analysis, here it has been chosen to better resolvethe smallest fluctuations by choosing �x�� and to use thesimple and computationally less expensive linear interpola-tion.
We stress that having well resolved dissipative physicsfor the Eulerian field is also very important for capturing theformation of rare structures on a scale r��. Moreover, asdiscussed in Ref. 53, such structures, because of their fila-mentary geometry, may influence not only viscous but alsoinertial range physics.
Another possible source of error comes from the loss ofaccuracy in the integration of Eq. �3� for very small veloci-ties due to round-off errors. This problem can be overcomeby adopting higher-order schemes for temporal discretiza-tion. For extremely high Reynolds numbers, it may also benecessary to use double precision arithmetic, while for mod-erate R�, single precision, which was adopted in the presentDNS, is sufficient for accurate results �see, e.g., Ref. 52�. Wealso remark that in our runs, round-off errors are alwayssubleading with respect to errors coming from interpolationor temporal discretization schemes.
Details of the DNS analyzed here can be foundelsewhere;23 here, we simply state that the Lagrangian trac-ers move according to Eq. �3�, in a cubic, triply periodicdomain of side B=2�. DNS parameters are summarized inTable II.
III. COMPARISON OF LAGRANGIAN STRUCTUREFUNCTIONS
Let us now compare the experimental and numericalmeasurements of the LVSFs directly. Figures 3�a� and 3�b�show a direct comparison of LVSFs of order p=2 and p=4
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.5 1 1.5 2 2.5 3 3.5 4
v’i2
(m/s
)
time (hours)(a)
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
v’i2
(a.u
.)
t (a.u.)
0.5
1
1.5
0 5 10 15 20
(b)
FIG. 2. �a� Time evolution of the components of the velocity fluctuation vx�2
�dashed line�, vy�2 �thick black line�, and vz�
2 �solid line� for EXP2. �b� Timeevolution of velocity fluctuations vi�
2, with i=x ,y ,z, for DNS2. In the inset,we show the same time evolution for a DNS at a smaller R��75 �obtainedwith a spatial resolution of 1283 grid points and the same forcing�, whichwas integrated for a much longer time. In the latter case, the three compo-nents fluctuate around the same value, showing the recovery of isotropy forlong enough time.
065103-5 Lagrangian structure functions in turbulence Phys. Fluids 20, 065103 �2008�
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for all data sets. The curves are plotted using the standardKolmogorov scaling, which assumes that in the inertialrange,
S2��� �� vrms�2 R�−1��/���
�where we have used the following relations: �vrms�3 /L �Ref.54� and TL /��R��. Such a formulation can be generalizedto Sp���vrms�p R�
−p/2�� /���p/2. Both the second- and fourth-order moments show a fairly good collapse, especially in therange of intermediate time lags. However, some dependencecan be observed both on R� �see Fig. 3�b�� and on the size ofthe measurement volume �compare EXP2 and EXP4�. Botheffects call for a more quantitative understanding.
A. Local scaling exponents
A common way to assess how the statistical propertieschange for varying time lags is to look at dimensionlessquantities such as the generalized flatness,
F2p��� =S2p���
�S2����p . �4�
We speak of intermittency when such a function changes itsbehavior as a function of �: This is equivalent to the PDF ofthe velocity fluctuations ��v, normalized to unit variance,changing shape for different �.34
When the generalized flatness varies with � as a powerlaw, F2p�������2p�, the scaling laws are intermittent. Suchbehavior is very difficult to assess quantitatively since manydecades of scaling are typically needed to remove the effectsof subleading contributions �for instance, it is known thatEulerian scaling may be strongly affected by slowly decay-ing anisotropic fluctuations56�.
We are interested in quantifying the degree of intermit-tency at changing �. In Fig. 4, we plot the generalized flat-ness F2p��� for p=2 and p=3 for the data sets DNS2, EXP2,and EXP4. Numerical and experimental results are veryclose and clearly show that the intermittency changes con-siderably going from small to large �.
TABLE II. Parameters of the numerical simulations. Taylor microscale Reynolds number R� 15vrms�2 / ��, root-mean-square velocity fluctuations vrms�=2E /3, where E is the kinetic energy, energy dissipation �, viscosity �, Kolmogorov length scale �= ��3 /��1/4, integral scale L equal to half side of thenumerical box, large-eddy turnover time TL=L /vrms� , Kolmogorov time scale ��= �� /��1/2, total integration time T, grid spacing �x, resolution N3, and thenumber of Lagrangian tracers Np. In the DNS, energy dissipation is measured as �=15����zvz�2�. Note that the values of vrms� and of T differ from thosereported in Ref. 23, where these values were misprinted.
No. R� vrms� � � � L TL �� T �x N3 Np
DNS1 178 1.4 0.886 0.002 05 0.01 3.14 2.2 0.048 5 0.012 5123 0.96�106
DNS2 284 1.4 0.81 0.000 88 0.0054 3.14 2.2 0.033 4.4 0.006 10243 1.92�106
10-2
10-1
100
101
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FIG. 3. �a� Log-log plot of the second-order structure function compensatedas R�S2��� /vrms�2 vs � /�� for all data sets at several Reynolds numbers. �b�The same for the fourth-order structure function R�
2S4��� /vrms�4 . The solid lineis made to guide the eyes through the two data sets �EXP2 and EXP4�obtained at the same Reynolds number in two different measurement vol-umes, as explained in Sec. II A.
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(τ)
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FIG. 4. Generalized flatness F2p��� of order p=2 and p=3 measured fromDNS2, EXP2, and EXP4. Data from EXP2 and EXP4 are connected by acontinuous line. The Gaussian values are given by the two horizontal lines.The curves have been averaged over the three velocity components and theerror bars �Ref. 55� shown here are computed from the scatter between thethree different components as a measure of the effect of anisotropy. Statis-tical errors due to the limitation in the statistics have been also evaluated bydividing the whole data sets in subsamples and comparing the results. Thesestatistical errors are always smaller than those plotted here, which wereestimated from the residual anisotropy.
065103-6 Biferale et al. Phys. Fluids 20, 065103 �2008�
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The difficulty in trying to characterize these changesquantitatively is that, as shown by Fig. 4, one needs to cap-ture variations over many orders of magnitude. For this rea-son, we prefer to look at observables that remain O�1� overthe entire range of scales and which convey informationabout intermittency without having to fit any scaling expo-nent. With this aim, we measured the logarithmic derivative�also called local slope or local exponent� of structure func-tion of order p, Sp���, with respect to a reference structurefunction,57 for which we chose the second-order S2���,
�p��� =d log�Sp����d log�S2����
. �5�
We stress the importance of taking the derivative with re-spect to a given moment: this is a direct way of looking atintermittency with no need of ad hoc fitting procedures andno request of power law behavior. This procedure,57 whichgoes under the name of extended self-similarity57 �ESS�, isparticularly important when assessing the statistical proper-ties at Reynolds numbers not too high and/or close to theviscous dissipative range.
A nonintermittent behavior would correspond to �p���= p /2. In the range of � for which the exponents �p��� aredifferent from the dimensional values p /2, structure func-tions are intermittent and correspondingly the normalizedPDFs of ���v / ����v�2�1/2� change shape with �. Figures 5�a�and 5�b� show the logarithmic local slopes of the numericaland experimental data sets for several Reynolds numbers forp=4 and p=6 versus time normalized to the Kolmogorovscale, � /��. These are the main results of our analysis.
The first observation is that for both orders p=4 and p=6, the local slopes �p��� deviate strongly from their nonin-termittent values �4=2 and �6=3. There is a tendency towardthe differentiable nonintermittent limit �p= p /2 only for verysmall time lags ����. In the following, we shall discuss indetail the small and larger time lag behavior, where by large,we mean �����TL.
Small time lags. For the structure function of order p=4 �Fig. 5�a��, we observe the strongest deviation from thenonintermittent value in the range of time 2�����6��. Ithas previously been proposed that this deviation is associatedwith particle trapping in vortex filaments.23 This fact hasbeen supported by DNS investigations of inertialparticles.14,17,29 The agreement between the DNS and the ex-perimental data in this range is remarkable. For p=6 �Fig.5�b��, the scatter among the data is higher due to the fact thatwith increasing order of the moments, inaccuracies in thedata become more important. Still, the agreement betweenDNS and the experimental data is excellent. Differently fromthe p=4 case, a dependence of mean quantities on the Rey-nolds number is here detectable, although it lies within theerror bars. The experimental data set for p=6, at the highestReynolds number �R�=815�, shows a detectable trend in thelocal slope toward less intermittent values in the dip region,2�� /���6. This change may potentially be the signature ofvortex destabilization at high Reynolds number—whichwould reduce the effect of vortex trapping. It is more likely,however, that at this very high Reynolds number, both spatial
and temporal resolution of the measurement system may nothave been sufficient to resolve the actual trajectories of in-tense events.23 We consider this to be an important openquestion for future studies.
Larger time lags. For � larger than �6–7��� up to TL, theexperimental data obtained in small measurement volumes�EXP1, 2, and 3� are not resolving the physics, as they de-velop both strong oscillations and a common trend towardsmaller and smaller values for the local slopes for increasing�. This may be attributed to the finite measurement volumeeffect �see also Sec. III B�. For these reasons, the data ofEXP1, 2, and 3 are not shown for these time ranges. On theother hand, the data from EXP4, obtained from a larger mea-surement volume, allow us to compare experiment and simu-lation. Here, the local slope of the experimental data changesslower very much akin to the simulations. This suggests thatin this region, high Reynolds number turbulence may show aplateau, although the current data cannot give a definitiveanswer to this question. For p=6, a similar trend is detected,although with larger uncertainties. The excellent quantitativeagreement between DNS and the experimental data gives us
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FIG. 5. Logarithmic derivatives �p��� of structure functions Sp��� with re-spect to S2��� for orders p=4 �a� and p=6 �b�. The curves are averaged overthe three velocity components and the error bars are computed from thestatistical �anisotropic� fluctuations between LVSFs of different components.The horizontal lines are the nonintermittent values for the logarithmic localslopes, i.e., �p= p /2. We stress that the curves for EXP1, 2, and 3 are shownin the time range 1�� /���7, while the curves for EXP4 �large measure-ment volume� are shown in the time range 7�� /���50.
065103-7 Lagrangian structure functions in turbulence Phys. Fluids 20, 065103 �2008�
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high confidence into the local slope behavior as a function oftime lag.
In light of these results, we can finally clarify the recentapparent discrepancy between measured scaling exponents ofthe LVSFs in experiments26 and DNS,23 which have lead tosome controversy in the literature.33,42,43 In the experimentalwork,26 scaling exponents were measured by fitting thecurves in Fig. 5 in the range 2�����6��, where the com-pensated second-order velocity structure functions reach amaximum, as shown in Fig. 1 �measuring the fourth- andsixth-order scaling exponents �p��� to be 1.4�0.1 and1.6�0.1, respectively�. On the other hand, in thesimulations,23 scaling exponents were measured in the re-gions in the range of times 10�����50�� �finding the val-ues �4=1.6�0.1 and �6=2.0�0.1�.
It needs to be emphasized, however, that the limits in-duced by the finiteness of volume and of the inertial rangeextension in both DNS and experimental data do not allowfor making a definitive statement about the behavior in theregion �10��. We may ask instead if the relative extensionof the interval where we see the large dip at ��2�� and thepossible plateau, observed for �10�� both in the numericaland experimental data �see EXP4 data set�, become larger orsmaller at increasing the Reynolds number.32 If the dipregion—the one presumably affected by vortex filaments—flattens, it would give the asymptotically stable scaling prop-erties of Lagrangian turbulence. If instead the apparent pla-teau region, at large times, increases in size while the effectof high intensity vortex remains limited to time lags around�2–6���, the plateau region would give the asymptotic scal-ing properties of Lagrangian turbulence. This point remains avery important question for the future because, as of today, itcannot be answered conclusively either by experiments or bysimulations.
B. Finite volume effects at large time lags
As noted above, the EXP4 data for �4��� develop anapparent plateau at a smaller value than the DNS data. In thissection, we show how the DNS data can be used to suggest apossible origin for this mismatch.
We investigate the behavior of the local slopes for thesimulations when the volume of size L3, where particles aretracked, is systematically decreased. We consider in theanalysis only trajectories which stay in subvolume of sizeL3, so partly mimicking what happens in the experimentalmeasurement volume. We consider volume sizes in the rangethat goes from the full box size with side B to boxes withside L=B /7, and we average over all the sub-boxes to in-crease the statistical samples. In Fig. 6, we plot the statisticsof the trajectory durations for both the experiment and DNS2by varying the measurement volume size.
As shown in Ref. 58, the probability of the trajectorydurations in finite volume measurements is independent ofthe size of the measurement volume if the trajectory durationis normalized by the residence time Tres=L /vrms� , which isthe case in Fig. 6�a� for DNS data. For the experimentaltrajectories, if the residence time is estimated from the geo-metrical side Bobs of the measurement volume as Tres
=Bobs /vrms� , the curve is off from others. This is explained byfact that experimental trajectories might still terminate pre-maturely for the reasons discussed in Sec. II A. Even after anattempt was made to connect trajectory segments, the effec-tive residence time could be smaller than that determinedfrom the geometrical size of the measurement volume. InFig. 6�a�, we show indeed that the probability for experimen-tal trajectories collapses with DNS curves, if we reduce theeffective side of the measurement box to � 2
3�Bobs or Tres
= � 23
�Bobs /vrms� . From the same plot, we might also notice thatthe DNS curve for L=B /2, which has the largest residentialtime Tres, slightly departs from the others. This is due tostatistics since those points that deviate belong to the tails ofthe distribution, where we do not expect a perfect collapse.
Finally, comparing Tres /�� then indicates that EXP2could be compared to DNS2 with subdomain of side B /4, asshown in Fig. 6�b�. Here, we are implicitly assuming thateven if particle track loss might have a different origin in theexperiment �optical and finite volume effects� and in the nu-merics �only finite volume effect�, the resulting statistics isbiased in a similar way. Also, we are assuming that Reynolds
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FIG. 6. �a� Comparison of the probability P�t /Tres� that a trajectory lasts atime t vs t /Tres for the experiment EXP2 and for DNS2 trajectories mea-sured in different numerical measurement domains L /B= 1
2 , 14 , 1
6 , 17 , where
Tres is the residence time and L=2� is the computational box. For DNStrajectories, Tres is determined from the size of the subdomain as Tres
=L /vrms� . For experimental trajectories, Tres= � 23
�Bobs /vrms� , where Bobs is thesize of the measurement volume, as given in Table I. Data for t /Tres2, forDNS at L /B= 1
4 , 16 , 1
7 and for the experiment, have been cut out. �b� Com-parison of the probability P�t /��� that a trajectory lasts a time t vs t /�� forthe same data as before.
065103-8 Biferale et al. Phys. Fluids 20, 065103 �2008�
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dependence is well accounted for by normalizing the resi-dence time with the Kolmogorov time scale. A deeper analy-sis of these issues is left to future work.
It is now interesting to look at the LVSF measured fromthese finite length numerical trajectories. In Fig. 7�a�, weshow the fourth-order LVSF obtained by considering the fulllength trajectories and the trajectories living in a subvolumeas explained above. What clearly appears from Fig. 7�a� isthat the finite length of the trajectories lowers the value ofthe structure functions for time lags of the order of 20��
���40��. Indeed, the finite-length statistics give a signalthat is always lower than the full averaged quantity: thiseffect may be due to a bias to slow, less energetic particles,which have a tendency to linger inside the volume for longertimes than fast particles, introducing a systematic change inthe statistics. Note that this is the same trend detected whencomparing EXP2 and EXP4 in Fig. 3. In Fig. 7�b�, we alsoshow the effect of the finite measurement volume on thelocal slope for p=4. By decreasing the observation volume,
we observe a trend in the local slope toward a shorter andshorter plateau with smaller and smaller values. In the samefigure, we also compare the logarithmic local slope of EXP2:the residence time analysis shows that EXP2 should be com-pared to DNS2 data in the subvolume with side L=B /4. Weobserve that at scales where both signals are present, thetrend is similar. However, we have to remember that in ex-periment, particles are lost when exiting the measurementvolume �finite volume effect� but also inside the measure-ment volume due to optical particle tracking limitations. Thismay explain the small discrepancy in Fig. 7�b�, among theEXP2 and the L=B /4 DNS2 signal. More generally, wethink that the loss of particles could be the source of thesmall offset between the plateaus developed by the EXP4data and the DNS data in Fig. 5.
For the sake of clarity, we should recall that in the DNS,particles can travel across a cubic fully periodic volume, soduring their full history they can re-enter the volume severaltimes. In principle, this may affect the results for long timedelays. However, since the particle velocity is taken at dif-ferent times, we may expect possible spurious correlationsinduced by the periodicity to be very small, if not absent.This is indeed confirmed in Fig. 7�b� where we can noticethe perfect agreement between data obtained by using peri-odic boundary conditions or limiting the analysis to subvol-umes of side L=B �i.e., not retaining the periodicity� andeven L=B /2.
C. Filtering and measurement error effectsat small time lags
As discussed in Sec. II, results at small time lags can beslightly contaminated by several effects both in DNS andexperiments. DNS data can be biased by resolution effectsdue to interpolation of the Eulerian velocity field at the par-ticle position. In experiments, uncorrelated experimentalnoise needs to be filtered to recover the trajectories.7,11,16
To understand the importance of such effects quantita-tively, we have modified the numerical Lagrangian trajecto-ries in the following way. First, we have introduced a randomnoise of the order of � /10 to the particle position in order tomimic the noise present in the experimental particle detec-tion. Second, we have implemented the same Gaussian filterof variable width used to smooth the experimental trajecto-ries x�t�. We also tested the effect of filtering by processingexperimental data with filters of different lengths.
In Figs. 8�a� and 8�b�, we show the local scaling expo-nents for �4���, as measured from these modified DNS tra-jectories together with the results obtained from the experi-ment, for several filter widths. The qualitative trend is verysimilar for both the DNS and the experiment. The noise inparticle position introduces nonmonotonic behavior in thelocal slopes at very small time lags in the DNS trajectories.This effect clearly indicates that small scale noise maystrongly perturb measurements at small time lags but will nothave important consequences for the behavior on time scaleslarger than ��. On the other hand, the effect of the filter is toslightly increase the smoothness at small time lags �noticethat the shift of local slopes curves toward the right for �
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FIG. 7. �a� The fourth-order structure function S4��� vs � /�� measured fromDNS2 trajectories for both full length trajectories �and with periodic bound-ary conditions� and for trajectories in smaller measurement volumes L /B= 1
4 , 17 . �b� The logarithmic local slope �4��� measured from DNS2 trajecto-
ries for both the full length trajectories �periodic boundary conditions� andfor trajectories in smaller measurement volumes L /B=1, 1
2 , 14 , 1
7 . Note thetendency toward a less developed plateau, at smaller and smaller values, asthe measurement volume decreases. In the same plot, we also compare thelocal slope of EXP2, whose trajectory length distribution is well reproducedby DNS2 data in the subvolume L /B= 1
4 .
065103-9 Lagrangian structure functions in turbulence Phys. Fluids 20, 065103 �2008�
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��� for increasing filter widths�. A similar trend is observedin the experimental data �Fig. 8�b��. In this case, choosingthe filter width to be in the range ��� 1
6 , 13��� seems to be
optimal, minimizing the dependence on the filter width andthe effects on the relevant time lags. Understanding filtereffects may be even more important for experiments withparticles much larger than the Kolmogorov scale. In thosecases, the particle size naturally introduces a filtering by av-eraging velocity fluctuations over its size, i.e., those particlesare not faithfully following the fluid trajectories.11,15
IV. CONCLUSION AND PERSPECTIVES
A detailed comparison between state-of-the-art experi-mental and numerical data of Lagrangian statistics in turbu-lent flows has been presented. The focus has been on single-particle Lagrangian structure functions. Only through thecritical comparison of experimental and DNS data is it pos-sible to achieve a quantitative understanding of the velocityscaling properties over the entire range of time scales and fora wide range of Reynolds numbers.
In particular, the availability of high Reynolds numberexperimental measurements allowed us to assess in a robust
way the existence of very intense fluctuations, with high in-termittency in the Lagrangian statistics around �� �2–6���.For larger time lags �10��, the signature of different sta-tistics seems to emerge, with again good agreement betweenDNS and experiment �see Fig. 5�. Whether the trend of loga-rithmic local slopes at larger times is becoming more andmore extended at larger and larger Reynolds number is anissue for further research.
Both experiments and numerics show in the ESS localslope of the fourth- and sixth-order Lagrangian structurefunctions a dip region at around time lags �2–6��� and aflattening at �10��. As of today, it is unclear whether thedip or the flattening region gives the asymptotic scalingproperties of Lagrangian turbulence. The question of whichregion will extend as a function of Reynolds number cannotbe resolved at present and remains open for future research.
It would also be important to probe the possible relationsbetween Eulerian and Lagrangian statistics, as suggested bysimple phenomenological multifractal models.13,23,46,47 Inthese models, the translation between Eulerian �single-time�spatial statistics and Lagrangian statistics is made via thedimensional expression of the local eddy turnover time atscale r: �r�r /�ru. This allows predictions for Lagrangianstatistics if the Eulerian counterpart is known. An interestingapplication concerns Lagrangian acceleration statistics,18
where this procedure has given excellent agreement with ex-perimental measurements. When applied to single-particlevelocities, multifractal predictions for the LVSF scaling ex-ponents are close to the plateau values observed in DNS attime lags �10��. It is not at all clear, however, if thisformalism is able to capture the complex behavior of thelocal scaling exponents close to the dip region �� �2–6���,as depicted in Fig. 5. Indeed, multifractal phenomenology, aswith all multiplicative random cascade models,34 does notcontain any signature of spatial structures such as vortex fila-ments. It is possible that in the Lagrangian framework, amore refined matching to the viscous dissipative scaling isneeded, as proposed in Ref. 13, rephrasing known results forEulerian statistics.40 Even less clear is the relevance for La-grangian turbulence of other phenomenological modelsbased on superstatistics,43 as recently questioned in Ref. 59.
The formulation of a stochastic model able to capture thewhole shape of local scaling properties from the smallest tolarger time lag, as depicted in Fig. 5, remains an open im-portant theoretical challenge.
ACKNOWLEDGMENTS
We thank an anonymous referee for suggesting us tonondimensionalize the trajectory length by the residencetime when plotting Fig. 6�a� and for pointing Ref. 58 to us.E.B., N.T.O., and H.X. gratefully acknowledge financial sup-port from the NSF under Contract Nos. PHY-9988755 andPHY-0216406 and by the Max Planck Society. L.B., M.C.,A.S.L., and F.T. acknowledge J. Bec, G. Boffetta, A. Celani,B. J. Devenish, and S. Musacchio for discussions and col-laboration in previous analysis of the numerical data set.L.B. acknowledges partial support from MIUR under theproject PRIN 2006. Numerical simulations were performed
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(b)
FIG. 8. �a� Logarithmic local slope �4��� for the DNS2 data set. The sym-bols DNS2a, b, and c denote the DNS2 trajectories modified by noise andfilter effects, mimicking the processing of experimental data. In particular,DNS2a refers to the introduction of noise in the particle position of the orderof �x�� /10 and with a Gaussian filter width ��� /3, DNS2b to the samefilter width but with much larger spatial noise ��x�� /4�, and DNS2c to thesame spatial noise but a large filter width �2�� /3. Note how when thefilter is not very large and with large spatial errors we have strong nonmono-tonic behavior for the local slopes �DNS2b�. �b� The effect of filter width ondata from EXP2 experiment �R�=690, small measurement volume�. Wetested four different filter widths: /��= 1
6 , 13 , 2
3 , and 43 .
065103-10 Biferale et al. Phys. Fluids 20, 065103 �2008�
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at CINECA �Italy� under the “key-project” grant: we thankG. Erbacci and C. Cavazzoni for resources allocation. L.B.,M.C., A.S.L., and F.T. thank the DEISA Consortium �co-funded by the EU, FP6 Project No. 508830� for supportwithin the DEISA Extreme Computing Initiative. Unproc-essed numerical data used in this study are freely availablefrom the iCFDdatabase.60
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